Solve PDE that has Boundary Conditions (different to wave equation) I don't have much knowledge in solving PDEs other than ones that I'm given in class. I was attempting a physics problem in which I came upon this PDE which is slightly different to the regular wave equation which is well known in how to solve:
$$\frac{\partial y}{\partial t} = \frac{1}{x^3}\frac{\partial}{\partial x}\left(x^3\frac{\partial y}{\partial x}\right).$$
I have these boundary conditions of $y (t = 0, x) = y_0$ and $y(t, x=X) = 0$. Could anyone assist me in finding a general solution for $y(t, x)$?
 A: Looks similar to the diffusion equation with spherical symmetry. We can separate variables: let
$$y(x,t)=X(x)T(t)$$
Substituting into the differential equation we find
$$\frac{T'}{T}=\frac{\partial_x(x^3X')}{x^3X}=\text{const}:=-k^2$$
Here $k$ is our separation constant. The equation for $T(t)$ is simple, yielding
$$
T(t)=A e^{-k^2t}
$$
Where the integration constant is $A$. The equation for $X(x)$ is
$$
-k^2x^2X=3xX'+x^2X''
$$ After the transformations $\zeta=X/x$ and $\chi=kx$, we find Bessel's equation of order one, so $\zeta=B J_1+C Y_1$ thus
$$
X(x)=x^{-1}\left(B J_1(kx)+C Y_1(kx) \right)
$$
Where $J_1$ is a Bessel function of the first kind and $Y_1$ is a Bessel function of the second kind and $B$ and $C$ are integration constants. Requiring $y$ be finite at the origin sets $C=0$, and the general solution is
$$
y(x,t)=\sum_k B_k e^{-k^2t} x^{-1} J_1(kx)
$$
Now the initial and boundary conditions. With $y(t,x_0)=0$ we find
$$
0=\sum_k B_k e^{-k^2t} x_0^{-1} J_1(kx_0)
$$
To hold for all $t$, we must have $J_1(kx_0)=0$, so $kx_0$ are roots of $J_1$. We can rewrite the sum with an integer index by denoting the $n$th root of $J_1$ as $\lambda_n$
$$
y(x,t)=\sum_{n=1}^\infty B_n e^{-\lambda_n^2t/x_0^2} x^{-1} J_1(\lambda_nx/x_0)
$$
The $B_n$ are found using the initial condition $y(x,0)=y_0(x)$
$$
y_0=\sum_{n=1}^\infty B_n  x^{-1} J_1(\lambda_nx/x_0)
$$
$$
x^2 y_0 =\sum_{n=1}^\infty B_n  x J_1(\lambda_nx/x_0)
$$
Exploiting the orthogonality relation of the Bessel functions we have
$$
B_n=\frac{2x_0}{J_2(\lambda_n)^2}\int\limits_0^1 d\eta \  \eta^2 J_1(\eta \lambda_n) y_0(\eta x_0)
$$
